Moufang sets arising from polarities of Moufang planes over octonion division algebras

Abstract

For every octonion division algebra $${\mathcal{O}}$$ , there exists a projective plane which is parametrized by $${\mathcal{O}}$$ ; these planes are related to rank two forms of linear algebraic groups of absolute type E 6. We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set. It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F 4, (2) Moufang sets of type 2 E 6, (3) hermitian Moufang sets of type C 4, and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra. Case (3) only occurs over fields of characteristic different from two, whereas case (4) only occurs over fields of characteristic equal to two. The Moufang sets of type 2 E 6 that we obtain in case (2) are exactly those corresponding to linear algebraic groups of type $${^2E^{29}_{6,1}}$$ ; the explicit description of those Moufang sets was not yet known.